Expression of type Implies

from the theory of proveit.logic.booleans.quantification.universality

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Function, P, n
from proveit.core_expr_types import P__x_1_to_n, P__x_1_to_np1, x_1_to_n, x_1_to_np1
from proveit.logic import And, Forall, Implies
from proveit.numbers import Natural

# build up the expression from sub-expressions
sub_expr1 = [n]
sub_expr2 = Forall(instance_param_or_params = [x_1_to_n], instance_expr = P__x_1_to_n)
expr = Implies(And(Function(P, []), Forall(instance_param_or_params = sub_expr1, instance_expr = Implies(sub_expr2, Forall(instance_param_or_params = [x_1_to_np1], instance_expr = P__x_1_to_np1)).with_wrapping_at(2), domain = Natural)), Forall(instance_param_or_params = sub_expr1, instance_expr = sub_expr2, domain = Natural)).with_wrapping_at(1)

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\begin{array}{c} \begin{array}{l} \left(P\left(\right) \land \left[\forall_{n \in \mathbb{N}}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{x_{1}, x_{2}, \ldots, x_{n}}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \Rightarrow  \\ \left[\forall_{x_{1}, x_{2}, \ldots, x_{n + 1}}~P\left(x_{1}, x_{2}, \ldots, x_{n + 1}\right)\right] \end{array} \end{array}\right)\right]\right) \\  \Rightarrow \left[\forall_{n \in \mathbb{N}}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n}}~P\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right]\right] \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(1)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 18 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 26 operand: 9
4	Literal
5	ExprTuple	7, 8
6	ExprTuple	9
7	Operation	operator: 33 operands: 10
8	Operation	operator: 26 operand: 13
9	Lambda	parameter: 44 body: 12
10	ExprTuple
11	ExprTuple	13
12	Conditional	value: 22 condition: 17
13	Lambda	parameter: 44 body: 15
14	ExprTuple	44
15	Conditional	value: 16 condition: 17
16	Operation	operator: 18 operands: 19
17	Operation	operator: 20 operands: 21
18	Literal
19	ExprTuple	22, 23
20	Literal
21	ExprTuple	44, 24
22	Operation	operator: 26 operand: 28
23	Operation	operator: 26 operand: 29
24	Literal
25	ExprTuple	28
26	Literal
27	ExprTuple	29
28	Lambda	parameters: 32 body: 30
29	Lambda	parameters: 34 body: 31
30	Operation	operator: 33 operands: 32
31	Operation	operator: 33 operands: 34
32	ExprTuple	35
33	Variable
34	ExprTuple	36
35	ExprRange	lambda_map: 37 start_index: 45 end_index: 44
36	ExprRange	lambda_map: 37 start_index: 45 end_index: 38
37	Lambda	parameter: 46 body: 39
38	Operation	operator: 40 operands: 41
39	IndexedVar	variable: 42 index: 46
40	Literal
41	ExprTuple	44, 45
42	Variable
43	ExprTuple	46
44	Variable
45	Literal
46	Variable